The dual nature of As-vacancies in LaFeAsO-derived superconductors: magnetic moment formation while preserving superconductivity
Konstantin Kikoin, Stefan-Ludwig Drechsler, Jiri Malek, Jeroen van den Brink
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t The dual nature of As-vacancies in LaFeAsO-derived superconductors:magnetic moment formation while preserving superconductivity
Konstantin Kikoin, Stefan-Ludwig Drechsler, ∗ Jiˇri M´alek,
2, 3 and Jeroen van den Brink School of Physics and Astronomy, Tel-Aviv University 69978 Tel-Aviv, Israel Institute for Theoretical Solid State Physics, IFW-Dresden, P.O. Box 270116, D-01171 Dresden, Germany Institute of Physics, ASCR, Prague, Czech Republic (Dated: May 4, 2017)As-vacancies (V As ) in La-1111-systems, which are nominally non-magnetic defects, are shown tocreate in their vicinity by symmetry ferromagnetically oriented local magnetic moments due to thestrong, covalent bonds with neighboring Fe atoms that they break. From microscopic theory in termsof an appropriately modified Anderson-Wolff model, we find that the moment formation results ina substantially enhanced paramagnetic susceptibility in both the normal and superconducting (SC)state. Despite the V As act as magnetic scatterers, they do not deteriorate SC properties which caneven be improved by V As by suppressing a competing or coexisting commensurate spin density waveor its remnant fluctuations. Due to the induced local moments an s ++ -scenario is unlikely. PACS numbers: 74.70.Xa, 74.25Ha, 71.55.Ak
Although the general features of the superconduct-ing (SC) and magnetic states in the Fe pnictides andchalcogenides are experimentally well-documented andtheoretical models convincingly outline a predominantlynon-phononic pairing, including multiband effects andunconventional pairing symmetries, the detailed mech-anism of Cooper pairing in these materials still awaitselucidation . From a general point of view, the studyand control of magnetic and non-magnetic defects canbe helpful in this respect since the way in which theyaffect the Cooper pairs depends directly on the pair-ing interactions and symmetries in these multiband sys-tems. In the pronounced multiband situation with un-conventional SC as considered in the Fe-pnictides, point-defects/impurities that affect the nonmagnetic interbandscattering will be detrimental for SC and is in no man-ner expected to strengthen it. Note several counterin-tuitive experimental observations related to the presenceof As-vacancies (V As ): (i) a significantly improved uppercritical field slope near a slightly enhanced T c but Pauli-limiting behavior above 30 T and a lacking thereof in”clean” samples , (ii) a strongly enhanced spin suscep-tibility pointing at the formation of magnetic momentslocalized in the vicinity of V As5 and (iii) a steepened de-scend of the NMR relaxation rate in the SC state ascompared with ”clean” non-deficient As systems. Herewe show that from a theoretical standpoint these obser-vations can be understood in terms of a highly unconven-tional role that V As play in magnetic and SC propertiesof the As-deficient, optimally doped LaFeAs − δ O − x F x .From the Anderson-Wolff model that we apply, itturns out that a nominally non-magnetic V As creates alocal magnetic moment due its breaking of four strong,covalent Fe-As bonds. This moment formation gives riseto the observed strongly enhanced paramagnetic suscep-tibility in both the normal and SC state. At the sametime, however, the presence of V As does not deteriorateSC properties and might even strengthen them.An V As in the anisotropic FeLaAs − x OF system can be considered as a dangling-bond (DB) defect in its Fe-As triple layers, which electronically can be viewed assingle layers (see Fig. 1). The V As , treated as a miss-ing As − , generates DB with 4 neighboring Fe ions inthe central square plane and 4 neighboring As ions intetrahedral coordination. In a minimal tight bindingmodel these bonds are formed by the p x,y and d xz,yz or-bitals. The corresponding dp and pp hopping integralsare denoted as W and W , respectively. This model re-flects the basic features of the band structure of LaFeAsO(see, e.g., ). According to density functional theorycombined with a tight binding analysis (DFT-TB), theFe-related 3 d bands range from -2 eV to +2 eV aroundthe Fermi level ε F = 0. The bonding and antibonding4 p x,y states contribute to As-related bands deep below ε F and the nearly empty bands above it. The partial densityof states (DOS) for these states is shown schematically (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ’ W x’y’ Γ M Γ (π,π) M X (π,0)(0,0) (0,π) xy W V As ββα FIG. 1: Bonds around a single As vacancy V As in a Fe-Aslayer. Fe ions: • . As ions above and below the Fe plane:solid and dashed ◦ , respectively. The dp -dangling bonds (DB)between the V As and its 4 Fe NN neighbors forming the orbital β are labeled as W ; pp -DB between V As orbital α and its 4neighbors in As tetrahedra are labeled as W . Inset: fragmentof the BZ for a single FeAs layer (dashed lines) and the foldedBZ for the Fe sublattice (full lines). The axes for full latticeand Fe sublattice are denoted as (x,y) and (x’,y’), respectively. in Fig. 2. The empty states in the very vicinity of ε F correspond to hole ( h ) and electron ( el ) pockets, and thepronounced step in the DOS at the upper band edgesarises due to the nearly 2D character of these dp -bands.In view of this structure of the energy spectrum, theexperimentally reported influence of V As defects on themagnetic properties of doped LaFeAsO looks especiallypuzzling. Indeed, the overlap between the p x,y statesand d yz,zx state is noticeable only near ε F . The mech-anism, which may be responsible for the formation oflocal magnetic moments in defect cells is the influenceof broken dp valence bonds described by the transferintegral W (Fig. 1) on the states near ε F . We con-clude from the geometry of the DB and the structure ofthe band spectrum that the V As − related defect potentialdisrupts mainly the hybridization with the d yz,xz partialcomponent of the d states and note that this transfer inte-gral should mainly contribute to the intraband scatteringwithin the h -pocket of the Fermi surface (FS), which isformed mainly by the d yz,xz states.The resulting Hamiltonian of this minimal model is H = H d + H p + H dp + H vd = X k σ ε k,d d † k σ d k σ + X k σ ε k ,p p † k σ p k σ + X k σ (cid:16) V dp ( k ) d † k σ p k σ + H . c . (cid:17) + W X jσ (cid:16) d † jσ p σ + H . c . (cid:17) . (1)The first three terms describe the nearly filled d xz,yz band ε k,d , the nearly empty p x,y band ε k,p , and the hybridiza-tion between them, respectively. H vd is the V As -inducedhybridization between the Wannier p -state in the defectcell labeled as ”0” and the d -band. An orbital splittingwill be irrelevant in what follows and we do not accountfor it here. It results in a double degeneracy of each band.The defect potential is given by the dp -DB characterizedby the coupling constant W βα = h β | W | α i (see Fig. 1).The energy spectrum of defect related states is deter-mined by the secular equation (see also Ref. 13)1 − X α W βα G αα ( ω ) W αβ G ββ ( ω ) = 0 , (2)The local Green’s functions (LGF) are definedas G αα ( ω ) = P k h α | k p ih k p | α i ω − ε k p and G ββ ( ω ) = P k h β | k d ih k d | β i ω − ε k d . The T -matrix for the scattering in the d -band is T kk ′ = F β ( k ) W β ( ω ) F β ( k ′ )1 − W β ( ω ) G ββ ( ω ) , with the structure factor F β ( k ) = h k d | β i and W β = P α W βα G αα ( ω ) W αβ .From general properties of the LGF S6 , we concludethat for repulsive vacancy potential W > U α ( ω ) is also positive for ω close to the topof the band ε a . Then we anticipate strong intrabandscattering in the h -pocket of the FS due to the nearly 2Delectronic structure. In this case the DOS at the top ε t ofthe h -band is constant ν , and this step-like singularity results in the logarithmic divergence of the LGF.Re G ββ ( ω → ε t ) ∝ ν ln [ | ω − ε t | /D ] . (3)Here D is an effective bandwidth (see e.g. Ref. S6). Suchan edge-singularity of the LGF means that the resonance(the zero in the denominator of the T -matrix should ap-pear near ε t , even if the scattering potential is weak (Fig.3, middle panel) so that in any case the impurity scat-tering in the h -pocket is close to the unitarity limit: thescattering phase δ ( ε F ) is close to π/
2. Thus, the DB in-duce a scattering in the ( xz, yz ) channel, which generatesmainly intraband scattering in the h -pocket.Now we turn to the magnetic structure of a [V As , Fe ]defect. The V As itself is not magnetically active, but in-volving its four Fe NN and possibly also four NNN inthe complex defect shown in Fig. 1 changes the situa-tion dramatically. Indeed, the necessary precondition forthe formation of localized magnetic states in a systemwith itinerant electrons is the presence of a noticeableshort-range spin-dependent interaction which may over-come the kinetic energy of the electrons. Fe ions boundwith V As may be the source of such interactions due tothe on-site Hubbard repulsion U in their 3 d shells. Tomodel this effect, one has to add the term U n β ↑ n β ↓ tothe Hamiltonian (1). Here U is the intracell Coulomb re-pulsion integral for the ”molecular orbital“ | β i (see Fig.1). We expect U b . U . U due to the slight delocaliza-tion of the 3 d wave functions because of dp -mixing anda reduced screening due to the missing V As10 , where U b denotes the Coulomb repulsion on bulk Fe-sites. Treat-ing this interaction in a mean-field manner results in anadditional spin-dependent term in the local scatteringpotential so that W β → W βσ = W β + U ¯ n β, − σ .One may expect that the broken dp -valence bonds inthe presence of a short range Coulomb repulsion in the Fe−d yz,zx −55010 0−2−4−6 2 DO S As−p x,y
E(eV) t ε E ε t FIG. 2: Partial DOS for Fe xz,yz and As x,y states (followingthe DFT-TB approximation ). E = 0 corresponds to ε F .Inset: vacancy induced peaks in the DOS near ε t (the top ofthe hole d , see also Fig. 3) d -shells of Fe ions involved in the formation of the de-fect can result in a spin-dependent scattering similarlyto the well known Wolff impurity model, S2,S1 which ex-plained the appearance of localized moments in metalswith potential scatterers in the same way as the Ander-son impurity model explained this effect in metals withresonant scatterers. A localized moment in a system ofitinerant electrons arises when the self-consistent solutionwith ¯ n β ↑ = ¯ n β ↓ of the Dyson equation for the electronGF exists, provided the repulsive potential exceeds somecritical value (see also Ref. 13). This means that twodefect-related narrow peaks arise in the DOS near ε F (see the inset in Fig. 2 and the lower panel in Fig. 3).In terms of the scattering phase shifts displayed in thelower panel of Fig. 3 the local magnetic order induced bythe defect [V As , Fe ] means that δ ↑ ( ε F ) > δ ↓ ( ε F ). Bothphases δ σ . π/
2, which means that the magnetic scat-tering is strong and not too far from the unitarity limit.Thus, we have found that the nominally nonmagneticV As defect can give rise to the appearance of localizedmoments formed by states in the h -pocket due to thequasi-2D character of the electronic band spectrum inferropnictides. This explains the observed strong en-hancement of the magnetic susceptibility χ (0) in an As-deficient La-1111 system , with an enhancement factor S1 of χ/χ p − ≈ cU χ p / (1 − U χ l ) , where χ p is the Pauli-spin susceptibility of the pristine La-1111 compound, c denotes the V As concentration, χ l = h S β , S β i is the localsusceptibility at the defect site.While the magnetic moment formation around V As sites explains the substantially enhanced paramagneticsusceptibility, one might expect that with respect to SC,it opens up a Pandora’s box. Why is the magneticmoment formation in the h -pockets not detrimental for εε ωωω −1 G π βββ W −1 W β δ π δ ω ω ε F tF
FIG. 3: Upper: DOS near ε t . Middle: Graphical solutionof Eq. U − ασ ( ω ) = Re G αα ( ω ) for spin ↑ (solid line) and spin ↓ (dashed line). Lower: Frequency dependent phase shifts δ ↑ ( ω ) and δ ↓ ( ω ) (solid and dashed curves, respectively). Theenergies ω σ mark the positions of resonances δ σ = π/ (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) σ p p σ = + p + Q, σ ’ (a) p σ (c)(b) FIG. 4: Diagrammatic representation for the anomalousvertex Γ (a), the susceptibility χ sdw responsible for theSDW fluctuations (b) and defect related corrections to χ sdw (c). Dashed and solid lines correspond to the el - and h -propagators, respectively. • are the anomalous Coulombvertices u (in notations of Ref. 35). Shaded circle stands forthe T -matrix (see the text for a more detailed explanation). SC ? The most straightforward explanation should be re-lated to cases when a competing or coexisting commen-surate stripe-like spin density wave (CS-SDW) phase orits short-range fluctuations are still present which detri-mental effect on SC is well-known. Then, the stronglyenhanced scattering of intinerant electrons from the h -pockets in their intraband channel found above, will helpto suppress further its influence and enhance T c . Thiseffect becomes weaker in the strongly overdoped regionwhere the SDW suppression by the doping itself is moreand more pronounced, Obviously, also the appearence ofrelatively large magnetic defects with ferromagneticallyordered large local moments provides a strong perturba-tion for such a CS-SDW found in undoped clean samples.As a result at low V As concentrations its transformationto an inhomogeneous magnetic state with discommensu-rations, or to a spin-glass type phase are expected. Forthem it is much easier to establish a coexistence of mag-netism and SC. The next intriguing puzzle is then whyis the formation of a magnetic moment in the h -pocketsnot detrimental for the SC as in usual s ++ -SC? The pos-sibility for a qualitatively new solution to this paradoxrelies on the observation that neither the standard theoryof doped single-band SC, nor the available approaches toimperfect multiband SC can be used to address thisissue in Fe-pnictide superconductors. Since the mecha-nism of SC in Fe-pnictides is not established as yet, westart with general remarks on the role of V As relateddefects in our SC for which actually also the symmetryof the order parameter is under debate, which may inprinciple be different in different Fe-based materials andeven depend on the doping type . To be definite,we will consider here only the most with a nodeless orderparameter ∆ sc having opposite signs for the h - and the el -pockets . In accordance with the theory of s ± multi-band SC in pnictides, the pairing is mainly given bythe anomalous vertices shown in Fig. 4 containing theinterband spin susceptibility χ SDW ( q , ω ) (Fig. 4b) as amain element. The spin density wave (SDW) fluctua-tions with the vector q = p + Q close to the Umklappvector connecting Γ and M in the BZ (Fig. 1) mediatethe Cooper pairing even in the absence (or suppression)of attractive interactions within el - and h -pockets of theBZ. This means that the main contribution of the mag-netic scattering to the h -propagator comes in via χ sdw (Fig. 4c), represented by the intraband T -matrix T pp . Itis seen from this diagram that we deal with magneticscattering without spin flips, which creates narrow localresonance levels below and above ε F in the h -pockets andmodifies the h -propagators in the bubble χ sdw ( q , ω ).Significantly SDW-affected Cooper pairing may berealized due to the almost singular behavior of χ sdw ( Q , ω ) ∝ ν ln[ D/max { ω, ǫ, γ } ], where D is the en-ergy interval where the nesting conditions are approxi-matily satisfied, ǫ and γ are parameters characterizingthe imperfection of nesting (including modifications ofthe magnetic response due to the presence of the V As ) andthe electron damping due to imperfection of a real crys-tal. The contribution of V As defects to ǫ are ∝ Re T ( ω ).Besides, As vacancies can act as dopants. The net re-sult of changed ǫ is not known a priori : based on theinterplay between these physical effects the nesting con-ditions may either slightly improve or worsen, or justremain unaltered. The damping γ . c | ε F − ω σ | is effi-cient only provided γ > ǫ . The main point is that eitherof these mechanisms cannot radically reduce the SC T c .Besides, magnetic resonances give their own contribution δχ sdw ( q , ω ) not related to the nesting. Transitions fromthe local states in the h- pocket to the empty states inthe el- pocket result in δχ sdw ( q , ω ) ∝ cν ln[ D/ ( ω −| ω σ | )].This contribution favors s ± -pairing. To summarize, froman analysis of the consequences of V As defects and theirmagnetic moment formation on the SC Fe-pnictides, itfollows that their presence can even be more construc-tive than detrimental for s ± SC that is mediated bySDW fluctuations. In this context it is interesting to notethe related conclusions on the role of intraband magneticscattering . The arguments given above are also in thespirit of the ”Swiss cheese” model , which presumes thatthe defect related bound states contribute to the subgapDOS and do not suppress T c completely. The spin susceptibility χ q ( ω ), is responsible also forthe 1 /T NMR spin-lattice relaxation rate, which changesfrom a nearly power-law dependence ∼ T in As stoichio-metric samples to more steep and close to an exponentialone in As-deficient samples. The analysis above suggeststhat these changes relate to the interplay between themid-gap states stemming from non-magnetic scatteringinduced by F doping and those inserted by magnetic de-fects V As (cf. Ref. 22). The interplay of magnetic vor-tices with both kinds of impurities, will affect the 1 /T rate . A detailed analysis will be published elsewhere.We have shown, in conclusion, that As-vacancies formhighly nontrivial defects in 1111 Fe-pnictide supercon-ductors which strongly modify their physical propertiesin both the normal and the SC states. Being nominallynon-magnetic in nature, they are nevertheless responsiblefor the formation of relatively large local magnetic mo-ments on the Fe-sites surrounding the vacancies, whichgive rise to enhanced spin susceptibility in the normalstate and Pauli-limiting behavior in the SC state. Thebehavior at low fields is unusual too, due to scatter-ing properties remarkably different from those of usualmagnetic impurities in standard single band and dirtymultiband s ± SC. Controling these defects can be help-ful to improve the understanding of real s ± systems, theelectronic structure, and correlation effects in the pnic-tides, in general. In particular, for 122-pnictides with Asvacancies and Fe-chalcogenides one expects a similarbut somewhat weaker scattering effect due to the largerelectronic dispersion perpendicular to the FeAs planes.An analysis on various types of point-defects will alsobe of considerable interest, since our results imply thatthey may strongly affect the physical properties of realpnictide materials even at small defect concentrations .We thank the DFG SPP 1458 and the Pakt f¨ur For-schung (IFW-Dresden) for support and appreciate dis-cussions with H. Eschrig † , M. Kiselev, K. Koepernik, D.Efremov, R. Kuzian, G. Fuchs, and J. Engelmann ∗ Electronic address: [email protected] D.C. Johnston, Adv. Phys. , 803 (2010). J. Paglione et al. , Nature Physics , 645 (2010). O.K. Andersen and L. Boeri, Annalen der Physik , 8(2011). G. Fuchs, et al , Phys. Rev. Lett. et al,
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Konstatin Kikoin , Stefan-Ludwig Drechsler , Jiˇri M´alek , , and Jeroen van den Brink School of Physics and Astronomy, Tel-Aviv University, 69978 Tel-Aviv, Israel IFW-Dresden, P.O. Box 270116, D-01171 Dresden, Germany Institute of Physics, ASCR, Prague, Czech RepublicIn the present Supplementary part we provide the reader with the details of the calculations of the [V As ,Fe ]-defectrelated Green’s function and the spectrum of local electron excitations. We present also an outlook about theinteraction of As-vacancies with an SDW or its fluctuations in a nonmagnetic state. Derivation of the secular equation for the defect related states
Here we calculate the local Green’s function of electrons in the Fe 3 d band containing a defect from four brokenvalence bonds with the 4 p bands related to the As sublattice. In the Bloch representation the defect related scatteringHamiltonian H vd reads as H vd = W X k σ h F ( k ) d † k σ p σ + H . c . i , (S1)where F ( k ) is the structure factor depending on the symmetry of the defect potential (see below). The scatteringterm H vd describes the perturbation inserted by a V As in the d -band, and our task is to calculate the reconstructionof the states in the hole pocket induced by this perturbation.The Green’s functions for the band Hamiltonian form the matrix G ( ω ): G ( ω ) = (cid:18) G dd ( ω ) 00 G pp ( ω ) (cid:19) (S2)(here the spin index is temporarily omitted). The perturbation ∼ W is nonzero in a limited space around an As-vacancy. The states within this cluster are described in a local basis ( α, β ) formed by projecting the states ( p, d ) ontothe perturbed region. Then the perturbation is described by the matrix W , W = (cid:18) W βα W αβ (cid:19) (S3)so that only the dp bonds are broken in the defect cluster.The local basis obeys the point symmetry of the 2D lattice. In the simplest tight-binding approximation the states | α i are the orbital states p x , p y centered at the As-vacancy site “0”. Then the states | β i are the “molecular” orbitalsformed by d yz,zx orbitals centered at the sites 1,2,3,4 surrounding the As-vacancy site “0” and transforming along thesame irreducible representation of the point group as the states | α i , namely the combinations | β i = | d i − | d i + | d i − | d i , where the Fe-sites in the first coordination sphere around the vacancy are enumerated as 1,2,3,4. Next we constructthe secular matrix Q ( ω ) = − WG ( ω ) and project this matrix on the local basis { α, β } , Q → e Q . e Q ( ω ) = (cid:18) − W αβ G ββ ( ω ) − W βα G αα ( ω ) 1 (cid:19) . (S4)We derive from (S4) the secular equation det e Q ( ω ) = 0. Since we are interested in the scattering in the band β relatedto the d -states, we project this secular equation on the subset h β | . . . | β i .1 − X α W βα G αα ( ω ) W αβ G ββ ( ω ) = 0 . (S5) W βα = h β | W | α i . Comparing Eq. (S5) with a Slater-Koster-like equation 1 − W G = 0 for a single band defectcharacterized by the local potential W we note that in our problem this potential is substituted for the non-localpotential W β : 1 − W β ( ω ) G ββ ( ω ) = 0 (S6) W β ( ω ) = X α W βα G αα ( ω ) W αβ . The solution of Eq. (S5) provides us with the information about the scattering phase δ β ( ω )tan δ β ( ω ) = − Im det e Q ( ω )Re det e Q ( ω ) . (S7)which characterizes the strength of the defect potential at the Fermi level ω = ε F . The T -matrix for the scattering inthe d -band is T kk ′ = F β ( k ) W β ( ω ) F β ( k ′ )1 − W β ( ω ) G ββ ( ω ) . (S8)with the structure factor F β ( k ) = h k d | β i . Due to the predominantly d -character of the hole band, it is convenient totreat the problem in the square lattice with a folded Brillouin zone (see Fig. 1 in the main text). Then the structurefactor is F β ( k ) ≈ k x / − cos k y / ε k p neglected in the above calculations. Thiscontribution would result in a modification of the local Green’s function G αβ ( ω ) in the effective potential (S6). Insteadof the form h α | G kk | α i used in the right hand side of Eq. (S8), one should project the Green’s function on the subset h β | . . . | β i with the Slater-Koster defect in this band, namely, change h α | G kk | α i → h α | G kk (cid:18) W G kk − W G αα (cid:19) | α i (S9)(see Fig. 1 in the main text for the definition of W ).The first term in the right hand side of Eq. (S9) is the Green’s function projected onto the local p -orbitals | α i .Corrections due to the contribution of dangling bond states in the p -band would be important only for those energieswhere 1 − W G αα ( ω ) ∼
0. However, it is seen from Eqs. (S6) and (S9) and from the shape of the DOS in Fig. 2 thatthe function G αα ( ω ) is smooth in the region of overlap with the partial d component of the DOS, and the singularitiesin the Green’s function reflecting the 2D van Hove singularities in the DOS are located around the top of the p bandin the region of unoccupied states. In other words, W G αα ( ω ) ≪ p wave in Eq.(S5) is represented by the unperturbed orbital | α i . Search for magnetic solutions
In order to find a magnetic solution, the Coulomb interaction is included in the scattering potential, W β → W βσ = W β + U ¯ n β, − σ . The average ¯ n βσ is given by¯ n βσ = 1 iπN X k k Im Z ε F F β ( k ) F β ( k ) G k k ,dσ ( ω ) dω = 1 iπ Z ε F G ββ,σ ( ω ) dω. . (S10)Its defect related part reads S1 ¯ n ′ βσ = 1 iπU βσ Z ε F Im " − W βσ G ββ,σ ( ω ) dω. (S11)It is known S1,S2 , that a magnetic solution n β ↑ = n β ↓ exists provided the repulsive potential exceeds some criticalvalue, W βc > / J c given by J c = 1 iπ Z ε F Im ( [ − G ββ ( ω )] [1 − W βc G ββ ( ω )] ) dω. (S12)Such a magnetic solution is expected to be realized due to the logarithmic singularity of G ββ ( ω ) in the very vicinityof ε F . shown in Eq. (5) in the main text. Due to this singularity in the denominator of the integrand in the r.h.s. ofEq. (S12) the factor J c is strongly enhanced. This enhancement favors magnetic solutions.The electron-band may be easily included in this calculation scheme, and the corresponding secular matrix may beconstructed in the same way as it was done above for the hole-band. In order to describe the states in the electronpockets (as well as the interband scattering), one should add at least one more orbital, namely the d xy one, tothis minimal model S3,S4,S5 . However, the real part of the local Green’s function G ( ω ) < S6 , so the intraband scattering in the electron pocket is expected to be weak. Thus, weconclude that a V As defect influences mainly the states in the hole pocket, leaving the electron pockets practicallythe same as in perfect samples. Few remarks on As-vacancies and spin-density wave states
Finally, we would like to mention an additional challenging problem closely related to that of As-vacancies in aparamagnetic enviroment considered above: namely, the perturbational effect of As-vacancies on a surrounding spindensity wave (SDW) to be investigated in more detail elsewhere. From a general point of view it is however alreadyclear that sizable effects of common interest can be expected. Indeed, since the competing spin-stripe SDW phasemay be schematically represented by a frustrated 2D J - J or closely related spin model Hamiltonians with an essen-tial antiferromagnetic next nearest neighbor exchange coupling J microscopically mediated by the superexchangeinvolving the As-4 p states, the presence of an As-vacancy will locally eliminate this antiferromagnetic coupling J even in the case of a nonmagnetic bound or resonance state discussed above. Moreover, in case that the SDW (orits by doping weakened corresponding SDW-magnetic state) will not prevent the ferromagnetic polarization effect forthe four sourrounding Fe sites, the presence of such a local ferromagnetic ”mini”-cluster as an extended magneticdefect will obviously cause an additional weakening or destruction of the SDW or its fluctuations probably presenteven in the ”non-magnetic” superconducting state mentioned above. This effect might explain the observed slight T c -enhancement of about 2 to 3 K after creating As-vacancies within an optimal doped system (see Ref. 4 of the maintext). Within a broader context magnetic defects under control and As-vacancies in particular should provide a newtool to probe various SDW states and this way give more insight into the complex interplay of various competingground states of Fe pnictides in general. ∗ Electronic address: [email protected] S1 D.J. Mills and P. Lederer, Phys. Rev. , 590 (1967). S2 P.A. Wolff, Phys. Rev. , 1030 (1961). S3 Y. Yanagi, et al , Phys. Rev. B , 054518 (2010). S4 S. Graser, et al.,
New J. Phys. , 025016 (2009). S5 T.A. Maier, et al , Phys. Rev. B , 224510 (2009). S6 E.N. Economou,